Third-power type

It is illuminating to compare the third-power type of equation for free chains and for tails. These both follow the general formula... 

This relation predicts that is asymptotically proportional to 2 for laige 2. Expressions for or(2) (and also as( ) which appears below) which give this type of asymptotic behavior are often called fifth-power type. Another feature of Flory s theory is that it permits evaluation of the proportionality constant (called the prefactor or front factor) in the asymptotic relation r 2. In a paper of 1960, Kurata et al. [15] criticized eq 1.15 by deriving an expression of third-power type on the basis of an ellipsoid polymer coil model. Although their conclusion was not ultimately accepted, this paper played an important role because it triggered intensive debates over the chain length dependence of polymer dimensions in the years to come. 

The dimensionless relations are usually indicated in either of two forms, each yielding identical resiilts. The preferred form is that suggested by Colburn ran.s. Am. In.st. Chem. Eng., 29, 174—210 (1933)]. It relates, primarily, three dimensionless groups the Stanton number h/cQ, the Prandtl number c Jk, and the Reynolds number DG/[L. For more accurate correlation of data (at Reynolds number <10,000), two additional dimensionless groups are used ratio of length to diameter L/D and ratio of viscosity at wall (or surface) temperature to viscosity at bulk temperature. Colburn showed that the product of the Stanton number and the two-thirds power of the Prandtl number (and, in addition, power functions of L/D and for Reynolds number <10,000) is approximately equal to half of the Fanning friction fac tor//2. This produc t is called the Colburn j factor. Since the Colburn type of equation relates heat transfer and fluid friction, it has greater utility than other expressions for the heat-transfer coefficient. 

However, scale-up of this type of system is limited due to the fact that oxygen transfer is a function of the area of the wave-containing liquid surface. Since the area increases with the square and the volume with the third power of a linear dimension, it is expected that this technology will reach a scale limit. Nevertheless, the companies that market this type of bioreactor offer bioreactors up to at least 500 L working volume (Wave Biotech, 2006). 

The length of the first quasi-stationary stage and relaxation time reaching the third equilibrium stage increases as a nontrivial power-type function of degrees of freedom N, namely x NlJ. This time scale is also confirmed by observing temporal evolution of single-particle distribution of momenta. The distributions taken at the same scaled time (i.e., t/NlJ = constant) are well-superposed irrespective of values of N, while a trivial time scale x N is numerically excluded. 

Agitated and Rotating Batch Dryers Scale up from pilot-plant tests in a small-scale dryer at the same temperature and pressure and similar agitation conditions. As noted under scoping design, scale-up depends on the surface area/volume ratio, and hence normally to the one-third power of mass. Results from one dryer type may be extrapolated to a different type if assumptions are made on the heat transfer coefficients in both dryers obviously this is less reliable than measurements on the same dryer type. 

Ne, the so-called Newton number, is tabulated in the standard literature for numerous types of agitators [58,59]. The revolution rate of the stirrer determines to the first power the conductive heat removal rate and to the third power the dissipated stirring power, respectively. 

Though these new theoretical values of 7 are clearly different from 5, the differences are only about 10%, indicating that the mean-field theory of Flory nearly hit the target. This means that eq 1.15 is by no means an unreasonable approximation for a discussion of excluded-volume effects in polymer solutions. The problem is whether 2.60 in it is adequate or not. It is also apparent that the various closed approximate equations of third or fourth-power type derived in the 1960s now have lost their significance. 

By contrast, Horner and Coleman get a third-power dependence on amine concentration for extractions of Pu(IV) from HgSO solutions by primary amines. They also determined distribution coefficients for Pu(IV) of several secondary and tertiary amines. Their results are shown in Fig. 29, showing the successive lowering of the distribution coefficients in going to more complex amine types. They report variable... 

Let us consider the simple case of two ions, each with one excitable electronic state separated from its electronic ground state by nearly equal energy. With suitable interaction between the two electronic systems, the excitation will jump from one ion to the other before a quantum of fluorescence is emitted. The systems interact by Coulomb interactions of the Van der Waals type. Forster (1948), who first treated such a case by quantum-mechanical theory, considered the dipole-dipole interactions. He assumed that the interaction is strongest if, for both transitions, electric-dipole transitions are allowed (Forster 1960). The interaction energy (//sa) is then proportional to the inverse of the third power of the interionic distance, and the transfer probability is given by... 

When a polymer is subject to an intense sinusoidal electric field such as that due to an intense laser pulse, Fourier analysis of the polarization response can be shown to contain not only terms in the original frequency co, but also terms in 2(0 and 3nonlinear response depends on the square of the intensity of the incident beam for 2co, and the third power for 3 . For the second-order effects, the system must have some asymmetry, as discussed previously. For poling, this means both high voltage and a chemical organization that will retain the resulting polarization for extended periods of time. Polymeric systems investigated have been of three basic types ... 

A third common type of gas laser is the carbon dioxide (COj) laser, which produces infrared light at 10.6 micrometers ( xm). The carbon dioxide laser is the most efficient and powerful of all gas lasers producing high powers-up to 1000 watts (W). Such a beam can easily cut through steel and is used in welding, drilling, and cutting. 

A termolecular elementary process in the gas phase involves a three-body collision, which we picture as a collision of a third particle with a pair of molecules that is undergoing a two-body collision. The number of three-body colUsions is proportional to the number of such pairs and is also proportional to the number of third particles. The rate of three-body collisions of a single substance is therefore proportional to the third power of the number density. The rate of three-body collisions of two particles of type 1 and a particle of type 2 is proportional to and so on. If we again... 

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